climxrrelem - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > climxrrelem		Structured version Visualization version GIF version

Theorem climxrrelem 46192

Description: If a sequence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

**Hypotheses**
Ref	Expression
climxrrelem.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
climxrrelem.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
climxrrelem.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
climxrrelem.c	⊢ (𝜑 → 𝐹 ⇝ 𝐴)
climxrrelem.d	⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
climxrrelem.p	⊢ ((𝜑 ∧ +∞ ∈ ℂ) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
climxrrelem.n	⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))

**Assertion**
Ref	Expression
climxrrelem	⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)

Distinct variable groups: 𝐴,𝑗 𝐷,𝑗 𝑗,𝐹 𝑗,𝑀 𝑗,𝑍 𝜑,𝑗

Proof of Theorem climxrrelem Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1921	. . . . 5 ⊢ Ⅎ𝑘𝜑
2		nfv 1921	. . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
3		nfra1 3263	. . . . . 6 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)
4	2, 3	nfan 1906	. . . . 5 ⊢ Ⅎ𝑘(𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
5	1, 4	nfan 1906	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
6		climxrrelem.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_≥‘𝑀)
7	6	uztrn2 12798	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
8	7	adantll 720	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
9		climxrrelem.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
10	9	fdmd 6665	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝑍)
11	10	ad2antrr 732	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → dom 𝐹 = 𝑍)
12	8, 11	eleqtrrd 2842	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
13	12	adantlrr 727	. . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
14		simpll 772	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝜑)
15	8	adantlrr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
16		rspa 3228	. . . . . . . 8 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
17	16	adantll 720	. . . . . . 7 ⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
18	17	adantll 720	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
19	9	ffvelcdmda 7025	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ^*)
20	19	3adant3 1138	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ∈ ℝ^*)
21		simpll 772	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → 𝜑)
22		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = -∞) → (𝐹‘𝑘) = -∞)
23		simpl 483	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = -∞) → (𝐹‘𝑘) ∈ ℂ)
24	22, 23	eqeltrrd 2840	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = -∞) → -∞ ∈ ℂ)
25	24	adantll 720	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → -∞ ∈ ℂ)
26		climxrrelem.n	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))
27	21, 25, 26	syl2anc 590	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))
28	27	adantlrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))
29		fvoveq1 7379	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = -∞ → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(-∞ − 𝐴)))
30	29	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = -∞) → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(-∞ − 𝐴)))
31		simpl 483	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = -∞) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)
32	30, 31	eqbrtrrd 5096	. . . . . . . . . . . . 13 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) < 𝐷)
33	32	adantll 720	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) < 𝐷)
34	33	adantlrl 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) < 𝐷)
35		climxrrelem.c	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
36	6	fvexi 6841	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑍 ∈ V
37	36	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑍 ∈ V)
38	9, 37	fexd 7171	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 ∈ V)
39		eqidd 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘))
40	38, 39	clim 15447	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))))
41	35, 40	mpbid 233	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
42	41	simpld 495	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℂ)
43	42	ad2antrr 732	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → 𝐴 ∈ ℂ)
44	25, 43	subcld 11496	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → (-∞ − 𝐴) ∈ ℂ)
45	44	abscld 15392	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
46	45	adantlrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
47		climxrrelem.d	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
48	47	rpred 12977	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ ℝ)
49	48	ad2antrr 732	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → 𝐷 ∈ ℝ)
50	46, 49	ltnled 11284	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → ((abs‘(-∞ − 𝐴)) < 𝐷 ↔ ¬ 𝐷 ≤ (abs‘(-∞ − 𝐴))))
51	34, 50	mpbid 233	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → ¬ 𝐷 ≤ (abs‘(-∞ − 𝐴)))
52	28, 51	pm2.65da 822	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = -∞)
53	52	3adant2 1137	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = -∞)
54	53	neqned 2941	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ≠ -∞)
55		simpll 772	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → 𝜑)
56		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = +∞) → (𝐹‘𝑘) = +∞)
57		simpl 483	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = +∞) → (𝐹‘𝑘) ∈ ℂ)
58	56, 57	eqeltrrd 2840	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = +∞) → +∞ ∈ ℂ)
59	58	adantll 720	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → +∞ ∈ ℂ)
60		climxrrelem.p	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
61	55, 59, 60	syl2anc 590	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
62	61	adantlrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
63		fvoveq1 7379	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = +∞ → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(+∞ − 𝐴)))
64	63	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = +∞) → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(+∞ − 𝐴)))
65		simpl 483	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = +∞) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)
66	64, 65	eqbrtrrd 5096	. . . . . . . . . . . . 13 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) < 𝐷)
67	66	adantll 720	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) < 𝐷)
68	67	adantlrl 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) < 𝐷)
69	42	ad2antrr 732	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → 𝐴 ∈ ℂ)
70	59, 69	subcld 11496	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → (+∞ − 𝐴) ∈ ℂ)
71	70	abscld 15392	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
72	71	adantlrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
73	48	ad2antrr 732	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → 𝐷 ∈ ℝ)
74	72, 73	ltnled 11284	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → ((abs‘(+∞ − 𝐴)) < 𝐷 ↔ ¬ 𝐷 ≤ (abs‘(+∞ − 𝐴))))
75	68, 74	mpbid 233	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → ¬ 𝐷 ≤ (abs‘(+∞ − 𝐴)))
76	62, 75	pm2.65da 822	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = +∞)
77	76	3adant2 1137	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = +∞)
78	77	neqned 2941	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ≠ +∞)
79	20, 54, 78	xrred 45809	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ∈ ℝ)
80	14, 15, 18, 79	syl3anc 1379	. . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ)
81	13, 80	jca 516	. . . 4 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))
82	5, 81	ralrimia 3238	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))
83	9	ffund 6659	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
84		ffvresb 7067	. . . . 5 ⊢ (Fun 𝐹 → ((𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)))
85	83, 84	syl 17	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)))
86	85	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) → ((𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)))
87	82, 86	mpbird 258	. 2 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) → (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)
88		breq2 5076	. . . . . 6 ⊢ (𝑥 = 𝐷 → ((abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
89	88	anbi2d 636	. . . . 5 ⊢ (𝑥 = 𝐷 → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
90	89	rexralbidv 3205	. . . 4 ⊢ (𝑥 = 𝐷 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
91	41	simprd 496	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
92	90, 91, 47	rspcdva 3561	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
93		climxrrelem.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
94	6	rexuz3 15302	. . . 4 ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
95	93, 94	syl 17	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
96	92, 95	mpbird 258	. 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
97	87, 96	reximddv 3155	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ∃wrex 3063 Vcvv 3431 class class class wbr 5072 dom cdm 5618 ↾ cres 5620 Fun wfun 6479 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 ℂcc 11027 ℝcr 11028 +∞cpnf 11167 -∞cmnf 11168 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 − cmin 11368 ℤcz 12515 ℤ_≥cuz 12779 ℝ⁺crp 12933 abscabs 15187 ⇝ cli 15437

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441

This theorem is referenced by: climxrre 46193

W3C validator